Abstract

This paper presents an efficient data-driven method to extract the effective metamaterial parameters unambiguously. This method is based on detecting the discontinuity points in the real part of the refractive index and discerning the correct branch values at these discontinuities. The proposed method is numerically simple and does not require an infinite frequency integration. The performance of the proposed method is demonstrated by investigating thin and thick homogenous and periodically structured metamaterial slabs. This method attains correct values of the branch index and, hence, correct values of the refractive index at the entire frequency range for thin and thick slabs independently from its imaginary part. It, notably, succeeds in overcoming the frequency and dimension limits and saturation observed in the Kramers–Kronig (K–K) method in thick homogeneous and multi-cell periodically structured slabs. The proposed method also gives accurate values for the metamaterial parameters at resonance and at the negative parameter region.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  3. X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70(1), 016608 (2004).
    [Crossref]
  4. Z. Szabó, G.-H. Park, R. Hedge, and E.-P. Li, “A unique extraction of metamaterial parameters based on Kramers–Kronig relationship,” IEEE Trans. Microwave Theory Tech. 58(10), 2646–2653 (2010).
    [Crossref]
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    [Crossref]
  8. V. Fokin, M. Ambati, C. Sun, and X. Zhang, “Method for retrieving effective properties of locally resonant acoustic metamaterials,” Phys. Rev. B 76(14), 144302 (2007).
    [Crossref]
  9. C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77(19), 195328 (2008).
    [Crossref]
  10. A. Starr, P. Rye, D. Smith, and S. Nemat-Nasser, “Fabrication and characterization of a negative-refractive-index composite metamaterial,” Phys. Rev. B 70(11), 113102 (2004).
    [Crossref]
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    [Crossref]
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    [Crossref]
  13. Y. Shi, T. Hao, L. Li, and C. H. Liang, “An improved NRW method to extract electromagnetic parameters of metamaterials,” Microw. Opt. Technol. Lett. 58(3), 647–652 (2016).
    [Crossref]
  14. K. Mnasri, A. Khrabustovskyi, M. Plum, and C. Rockstuhl, “Retrieving effective material parameters of metamaterials characterized by nonlocal constitutive relations,” Phys. Rev. B 99(3), 035442 (2019).
    [Crossref]
  15. G. T. Papadakis, P. Yeh, and H. A. Atwater, “Retrieval of material parameters for uniaxial metamaterials,” Phys. Rev. B 91(15), 155406 (2015).
    [Crossref]
  16. T. Koschny, P. Markoš, D. Smith, and C. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68(6), 065602 (2003).
    [Crossref]
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    [Crossref]
  19. U. C. Hasar, J. J. Barroso, C. Sabah, and Y. Kaya, “Resolving phase ambiguity in the inverse problem of reflection-only measurement methods,” Prog. Electromagn. Res. 129, 405–420 (2012).
    [Crossref]
  20. Y. Shi, Z.-Y. Li, L. Li, and C.-H. Liang, “An electromagnetic parameters extraction method for metamaterials based on phase unwrapping technique,” Waves in Random and Complex Media 26(4), 417–433 (2016).
    [Crossref]
  21. Z. Cao, F. Yuan, and L. Li, “An automated phase correction algorithm for retrieving permittivity and permeability of electromagnetic metamaterials,” AIP Adv. 4(6), 067115 (2014).
    [Crossref]
  22. G. Lu, Z. Duan, H. Yin, Z. Xiao, and J. Zhang, “Determining the Effective Electromagnetic Parameters of Photonic Crystal by Phase Unwrapping and Denoising Method,” Int. J. Antenn. Propag. 2019, 1–10 (2019).
    [Crossref]
  23. B. Fu, X. Ma, and G. Wan, “Retrieving the Constitutive Parameters of Metal Backed Radar Absorbing Material by Phase Unwrapping Method,” European Conference on Antennas and Propagation, 2018.
  24. P. Markoš and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express 11(7), 649–661 (2003).
    [Crossref]
  25. F.-J. Hsieh and W.-C. Wang, “Full extraction methods to retrieve effective refractive index and parameters of a bianisotropic metamaterial based on material dispersion models,” J. Appl. Phys. 112(6), 064907 (2012).
    [Crossref]
  26. N. Katsarakis, T. Koschny, M. Kafesaki, E. Economou, E. Ozbay, and C. Soukoulis, “Left-and right-handed transmission peaks near the magnetic resonance frequency in composite metamaterials,” Phys. Rev. B 70(20), 201101 (2004).
    [Crossref]
  27. R. Penciu, K. Aydin, M. Kafesaki, T. Koschny, E. Ozbay, E. Economou, and C. Soukoulis, “Multi-gap individual and coupled split-ring resonator structures,” Opt. Express 16(22), 18131–18144 (2008).
    [Crossref]

2019 (2)

K. Mnasri, A. Khrabustovskyi, M. Plum, and C. Rockstuhl, “Retrieving effective material parameters of metamaterials characterized by nonlocal constitutive relations,” Phys. Rev. B 99(3), 035442 (2019).
[Crossref]

G. Lu, Z. Duan, H. Yin, Z. Xiao, and J. Zhang, “Determining the Effective Electromagnetic Parameters of Photonic Crystal by Phase Unwrapping and Denoising Method,” Int. J. Antenn. Propag. 2019, 1–10 (2019).
[Crossref]

2017 (1)

Y. Shi, Z.-Y. Li, K. Li, L. Li, and C.-H. Liang, “A retrieval method of effective electromagnetic parameters for inhomogeneous metamaterials,” IEEE Trans. Microwave Theory Tech. 65(4), 1160–1178 (2017).
[Crossref]

2016 (2)

Y. Shi, T. Hao, L. Li, and C. H. Liang, “An improved NRW method to extract electromagnetic parameters of metamaterials,” Microw. Opt. Technol. Lett. 58(3), 647–652 (2016).
[Crossref]

Y. Shi, Z.-Y. Li, L. Li, and C.-H. Liang, “An electromagnetic parameters extraction method for metamaterials based on phase unwrapping technique,” Waves in Random and Complex Media 26(4), 417–433 (2016).
[Crossref]

2015 (1)

G. T. Papadakis, P. Yeh, and H. A. Atwater, “Retrieval of material parameters for uniaxial metamaterials,” Phys. Rev. B 91(15), 155406 (2015).
[Crossref]

2014 (1)

Z. Cao, F. Yuan, and L. Li, “An automated phase correction algorithm for retrieving permittivity and permeability of electromagnetic metamaterials,” AIP Adv. 4(6), 067115 (2014).
[Crossref]

2012 (2)

F.-J. Hsieh and W.-C. Wang, “Full extraction methods to retrieve effective refractive index and parameters of a bianisotropic metamaterial based on material dispersion models,” J. Appl. Phys. 112(6), 064907 (2012).
[Crossref]

U. C. Hasar, J. J. Barroso, C. Sabah, and Y. Kaya, “Resolving phase ambiguity in the inverse problem of reflection-only measurement methods,” Prog. Electromagn. Res. 129, 405–420 (2012).
[Crossref]

2010 (1)

Z. Szabó, G.-H. Park, R. Hedge, and E.-P. Li, “A unique extraction of metamaterial parameters based on Kramers–Kronig relationship,” IEEE Trans. Microwave Theory Tech. 58(10), 2646–2653 (2010).
[Crossref]

2008 (3)

2007 (2)

V. V. Varadan and R. Ro, “Unique retrieval of complex permittivity and permeability of dispersive materials from reflection and transmitted fields by enforcing causality,” IEEE Trans. Microwave Theory Tech. 55(10), 2224–2230 (2007).
[Crossref]

V. Fokin, M. Ambati, C. Sun, and X. Zhang, “Method for retrieving effective properties of locally resonant acoustic metamaterials,” Phys. Rev. B 76(14), 144302 (2007).
[Crossref]

2006 (1)

2005 (2)

D. Smith, D. Vier, T. Koschny, and C. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005).
[Crossref]

X. Chen, B.-I. Wu, J. A. Kong, and T. M. Grzegorczyk, “Retrieval of the effective constitutive parameters of bianisotropic metamaterials,” Phys. Rev. E 71(4), 046610 (2005).
[Crossref]

2004 (4)

D. R. Smith, J. B. Pendry, and M. C. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[Crossref]

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70(1), 016608 (2004).
[Crossref]

A. Starr, P. Rye, D. Smith, and S. Nemat-Nasser, “Fabrication and characterization of a negative-refractive-index composite metamaterial,” Phys. Rev. B 70(11), 113102 (2004).
[Crossref]

N. Katsarakis, T. Koschny, M. Kafesaki, E. Economou, E. Ozbay, and C. Soukoulis, “Left-and right-handed transmission peaks near the magnetic resonance frequency in composite metamaterials,” Phys. Rev. B 70(20), 201101 (2004).
[Crossref]

2003 (2)

P. Markoš and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express 11(7), 649–661 (2003).
[Crossref]

T. Koschny, P. Markoš, D. Smith, and C. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68(6), 065602 (2003).
[Crossref]

2002 (1)

D. R. Smith, S. Schultz, P. Markoš, and C. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

Ambati, M.

V. Fokin, M. Ambati, C. Sun, and X. Zhang, “Method for retrieving effective properties of locally resonant acoustic metamaterials,” Phys. Rev. B 76(14), 144302 (2007).
[Crossref]

Atwater, H. A.

G. T. Papadakis, P. Yeh, and H. A. Atwater, “Retrieval of material parameters for uniaxial metamaterials,” Phys. Rev. B 91(15), 155406 (2015).
[Crossref]

Aydin, K.

Barroso, J. J.

U. C. Hasar, J. J. Barroso, C. Sabah, and Y. Kaya, “Resolving phase ambiguity in the inverse problem of reflection-only measurement methods,” Prog. Electromagn. Res. 129, 405–420 (2012).
[Crossref]

Cao, Z.

Z. Cao, F. Yuan, and L. Li, “An automated phase correction algorithm for retrieving permittivity and permeability of electromagnetic metamaterials,” AIP Adv. 4(6), 067115 (2014).
[Crossref]

Chen, X.

X. Chen, B.-I. Wu, J. A. Kong, and T. M. Grzegorczyk, “Retrieval of the effective constitutive parameters of bianisotropic metamaterials,” Phys. Rev. E 71(4), 046610 (2005).
[Crossref]

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70(1), 016608 (2004).
[Crossref]

Dolling, G.

Duan, Z.

G. Lu, Z. Duan, H. Yin, Z. Xiao, and J. Zhang, “Determining the Effective Electromagnetic Parameters of Photonic Crystal by Phase Unwrapping and Denoising Method,” Int. J. Antenn. Propag. 2019, 1–10 (2019).
[Crossref]

Economou, E.

R. Penciu, K. Aydin, M. Kafesaki, T. Koschny, E. Ozbay, E. Economou, and C. Soukoulis, “Multi-gap individual and coupled split-ring resonator structures,” Opt. Express 16(22), 18131–18144 (2008).
[Crossref]

N. Katsarakis, T. Koschny, M. Kafesaki, E. Economou, E. Ozbay, and C. Soukoulis, “Left-and right-handed transmission peaks near the magnetic resonance frequency in composite metamaterials,” Phys. Rev. B 70(20), 201101 (2004).
[Crossref]

Enkrich, C.

Fokin, V.

V. Fokin, M. Ambati, C. Sun, and X. Zhang, “Method for retrieving effective properties of locally resonant acoustic metamaterials,” Phys. Rev. B 76(14), 144302 (2007).
[Crossref]

Fu, B.

B. Fu, X. Ma, and G. Wan, “Retrieving the Constitutive Parameters of Metal Backed Radar Absorbing Material by Phase Unwrapping Method,” European Conference on Antennas and Propagation, 2018.

Grzegorczyk, T. M.

X. Chen, B.-I. Wu, J. A. Kong, and T. M. Grzegorczyk, “Retrieval of the effective constitutive parameters of bianisotropic metamaterials,” Phys. Rev. E 71(4), 046610 (2005).
[Crossref]

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70(1), 016608 (2004).
[Crossref]

Hao, T.

Y. Shi, T. Hao, L. Li, and C. H. Liang, “An improved NRW method to extract electromagnetic parameters of metamaterials,” Microw. Opt. Technol. Lett. 58(3), 647–652 (2016).
[Crossref]

Hasar, U. C.

U. C. Hasar, J. J. Barroso, C. Sabah, and Y. Kaya, “Resolving phase ambiguity in the inverse problem of reflection-only measurement methods,” Prog. Electromagn. Res. 129, 405–420 (2012).
[Crossref]

Hedge, R.

Z. Szabó, G.-H. Park, R. Hedge, and E.-P. Li, “A unique extraction of metamaterial parameters based on Kramers–Kronig relationship,” IEEE Trans. Microwave Theory Tech. 58(10), 2646–2653 (2010).
[Crossref]

Hsieh, F.-J.

F.-J. Hsieh and W.-C. Wang, “Full extraction methods to retrieve effective refractive index and parameters of a bianisotropic metamaterial based on material dispersion models,” J. Appl. Phys. 112(6), 064907 (2012).
[Crossref]

Kafesaki, M.

R. Penciu, K. Aydin, M. Kafesaki, T. Koschny, E. Ozbay, E. Economou, and C. Soukoulis, “Multi-gap individual and coupled split-ring resonator structures,” Opt. Express 16(22), 18131–18144 (2008).
[Crossref]

N. Katsarakis, T. Koschny, M. Kafesaki, E. Economou, E. Ozbay, and C. Soukoulis, “Left-and right-handed transmission peaks near the magnetic resonance frequency in composite metamaterials,” Phys. Rev. B 70(20), 201101 (2004).
[Crossref]

Katsarakis, N.

N. Katsarakis, T. Koschny, M. Kafesaki, E. Economou, E. Ozbay, and C. Soukoulis, “Left-and right-handed transmission peaks near the magnetic resonance frequency in composite metamaterials,” Phys. Rev. B 70(20), 201101 (2004).
[Crossref]

Kaya, Y.

U. C. Hasar, J. J. Barroso, C. Sabah, and Y. Kaya, “Resolving phase ambiguity in the inverse problem of reflection-only measurement methods,” Prog. Electromagn. Res. 129, 405–420 (2012).
[Crossref]

Khrabustovskyi, A.

K. Mnasri, A. Khrabustovskyi, M. Plum, and C. Rockstuhl, “Retrieving effective material parameters of metamaterials characterized by nonlocal constitutive relations,” Phys. Rev. B 99(3), 035442 (2019).
[Crossref]

Kildishev, A. V.

Kong, J. A.

X. Chen, B.-I. Wu, J. A. Kong, and T. M. Grzegorczyk, “Retrieval of the effective constitutive parameters of bianisotropic metamaterials,” Phys. Rev. E 71(4), 046610 (2005).
[Crossref]

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70(1), 016608 (2004).
[Crossref]

Koschny, T.

R. Penciu, K. Aydin, M. Kafesaki, T. Koschny, E. Ozbay, E. Economou, and C. Soukoulis, “Multi-gap individual and coupled split-ring resonator structures,” Opt. Express 16(22), 18131–18144 (2008).
[Crossref]

D. Smith, D. Vier, T. Koschny, and C. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005).
[Crossref]

N. Katsarakis, T. Koschny, M. Kafesaki, E. Economou, E. Ozbay, and C. Soukoulis, “Left-and right-handed transmission peaks near the magnetic resonance frequency in composite metamaterials,” Phys. Rev. B 70(20), 201101 (2004).
[Crossref]

T. Koschny, P. Markoš, D. Smith, and C. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68(6), 065602 (2003).
[Crossref]

Kwon, D.-H.

Lederer, F.

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77(19), 195328 (2008).
[Crossref]

Li, E.-P.

Z. Szabó, G.-H. Park, R. Hedge, and E.-P. Li, “A unique extraction of metamaterial parameters based on Kramers–Kronig relationship,” IEEE Trans. Microwave Theory Tech. 58(10), 2646–2653 (2010).
[Crossref]

Li, K.

Y. Shi, Z.-Y. Li, K. Li, L. Li, and C.-H. Liang, “A retrieval method of effective electromagnetic parameters for inhomogeneous metamaterials,” IEEE Trans. Microwave Theory Tech. 65(4), 1160–1178 (2017).
[Crossref]

Li, L.

Y. Shi, Z.-Y. Li, K. Li, L. Li, and C.-H. Liang, “A retrieval method of effective electromagnetic parameters for inhomogeneous metamaterials,” IEEE Trans. Microwave Theory Tech. 65(4), 1160–1178 (2017).
[Crossref]

Y. Shi, T. Hao, L. Li, and C. H. Liang, “An improved NRW method to extract electromagnetic parameters of metamaterials,” Microw. Opt. Technol. Lett. 58(3), 647–652 (2016).
[Crossref]

Y. Shi, Z.-Y. Li, L. Li, and C.-H. Liang, “An electromagnetic parameters extraction method for metamaterials based on phase unwrapping technique,” Waves in Random and Complex Media 26(4), 417–433 (2016).
[Crossref]

Z. Cao, F. Yuan, and L. Li, “An automated phase correction algorithm for retrieving permittivity and permeability of electromagnetic metamaterials,” AIP Adv. 4(6), 067115 (2014).
[Crossref]

Li, Z.-Y.

Y. Shi, Z.-Y. Li, K. Li, L. Li, and C.-H. Liang, “A retrieval method of effective electromagnetic parameters for inhomogeneous metamaterials,” IEEE Trans. Microwave Theory Tech. 65(4), 1160–1178 (2017).
[Crossref]

Y. Shi, Z.-Y. Li, L. Li, and C.-H. Liang, “An electromagnetic parameters extraction method for metamaterials based on phase unwrapping technique,” Waves in Random and Complex Media 26(4), 417–433 (2016).
[Crossref]

Liang, C. H.

Y. Shi, T. Hao, L. Li, and C. H. Liang, “An improved NRW method to extract electromagnetic parameters of metamaterials,” Microw. Opt. Technol. Lett. 58(3), 647–652 (2016).
[Crossref]

Liang, C.-H.

Y. Shi, Z.-Y. Li, K. Li, L. Li, and C.-H. Liang, “A retrieval method of effective electromagnetic parameters for inhomogeneous metamaterials,” IEEE Trans. Microwave Theory Tech. 65(4), 1160–1178 (2017).
[Crossref]

Y. Shi, Z.-Y. Li, L. Li, and C.-H. Liang, “An electromagnetic parameters extraction method for metamaterials based on phase unwrapping technique,” Waves in Random and Complex Media 26(4), 417–433 (2016).
[Crossref]

Linden, S.

Lu, G.

G. Lu, Z. Duan, H. Yin, Z. Xiao, and J. Zhang, “Determining the Effective Electromagnetic Parameters of Photonic Crystal by Phase Unwrapping and Denoising Method,” Int. J. Antenn. Propag. 2019, 1–10 (2019).
[Crossref]

Lucarini, V.

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers–Kronig Relations in Optical Materials Research, vol. 110 (Springer Science & Business Media, 2005).

Ma, X.

B. Fu, X. Ma, and G. Wan, “Retrieving the Constitutive Parameters of Metal Backed Radar Absorbing Material by Phase Unwrapping Method,” European Conference on Antennas and Propagation, 2018.

Markoš, P.

T. Koschny, P. Markoš, D. Smith, and C. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68(6), 065602 (2003).
[Crossref]

P. Markoš and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express 11(7), 649–661 (2003).
[Crossref]

D. R. Smith, S. Schultz, P. Markoš, and C. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

Menzel, C.

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77(19), 195328 (2008).
[Crossref]

Mnasri, K.

K. Mnasri, A. Khrabustovskyi, M. Plum, and C. Rockstuhl, “Retrieving effective material parameters of metamaterials characterized by nonlocal constitutive relations,” Phys. Rev. B 99(3), 035442 (2019).
[Crossref]

Nemat-Nasser, S.

A. Starr, P. Rye, D. Smith, and S. Nemat-Nasser, “Fabrication and characterization of a negative-refractive-index composite metamaterial,” Phys. Rev. B 70(11), 113102 (2004).
[Crossref]

Ozbay, E.

R. Penciu, K. Aydin, M. Kafesaki, T. Koschny, E. Ozbay, E. Economou, and C. Soukoulis, “Multi-gap individual and coupled split-ring resonator structures,” Opt. Express 16(22), 18131–18144 (2008).
[Crossref]

N. Katsarakis, T. Koschny, M. Kafesaki, E. Economou, E. Ozbay, and C. Soukoulis, “Left-and right-handed transmission peaks near the magnetic resonance frequency in composite metamaterials,” Phys. Rev. B 70(20), 201101 (2004).
[Crossref]

Pacheco, J.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70(1), 016608 (2004).
[Crossref]

Papadakis, G. T.

G. T. Papadakis, P. Yeh, and H. A. Atwater, “Retrieval of material parameters for uniaxial metamaterials,” Phys. Rev. B 91(15), 155406 (2015).
[Crossref]

Park, G.-H.

Z. Szabó, G.-H. Park, R. Hedge, and E.-P. Li, “A unique extraction of metamaterial parameters based on Kramers–Kronig relationship,” IEEE Trans. Microwave Theory Tech. 58(10), 2646–2653 (2010).
[Crossref]

Paul, T.

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77(19), 195328 (2008).
[Crossref]

Peiponen, K.-E.

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers–Kronig Relations in Optical Materials Research, vol. 110 (Springer Science & Business Media, 2005).

Penciu, R.

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[Crossref]

Pertsch, T.

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77(19), 195328 (2008).
[Crossref]

Plum, M.

K. Mnasri, A. Khrabustovskyi, M. Plum, and C. Rockstuhl, “Retrieving effective material parameters of metamaterials characterized by nonlocal constitutive relations,” Phys. Rev. B 99(3), 035442 (2019).
[Crossref]

Ro, R.

V. V. Varadan and R. Ro, “Unique retrieval of complex permittivity and permeability of dispersive materials from reflection and transmitted fields by enforcing causality,” IEEE Trans. Microwave Theory Tech. 55(10), 2224–2230 (2007).
[Crossref]

Rockstuhl, C.

K. Mnasri, A. Khrabustovskyi, M. Plum, and C. Rockstuhl, “Retrieving effective material parameters of metamaterials characterized by nonlocal constitutive relations,” Phys. Rev. B 99(3), 035442 (2019).
[Crossref]

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77(19), 195328 (2008).
[Crossref]

Rye, P.

A. Starr, P. Rye, D. Smith, and S. Nemat-Nasser, “Fabrication and characterization of a negative-refractive-index composite metamaterial,” Phys. Rev. B 70(11), 113102 (2004).
[Crossref]

Saarinen, J. J.

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers–Kronig Relations in Optical Materials Research, vol. 110 (Springer Science & Business Media, 2005).

Sabah, C.

U. C. Hasar, J. J. Barroso, C. Sabah, and Y. Kaya, “Resolving phase ambiguity in the inverse problem of reflection-only measurement methods,” Prog. Electromagn. Res. 129, 405–420 (2012).
[Crossref]

Schultz, S.

D. R. Smith, S. Schultz, P. Markoš, and C. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

Shalaev, V. M.

Shi, Y.

Y. Shi, Z.-Y. Li, K. Li, L. Li, and C.-H. Liang, “A retrieval method of effective electromagnetic parameters for inhomogeneous metamaterials,” IEEE Trans. Microwave Theory Tech. 65(4), 1160–1178 (2017).
[Crossref]

Y. Shi, T. Hao, L. Li, and C. H. Liang, “An improved NRW method to extract electromagnetic parameters of metamaterials,” Microw. Opt. Technol. Lett. 58(3), 647–652 (2016).
[Crossref]

Y. Shi, Z.-Y. Li, L. Li, and C.-H. Liang, “An electromagnetic parameters extraction method for metamaterials based on phase unwrapping technique,” Waves in Random and Complex Media 26(4), 417–433 (2016).
[Crossref]

Smith, D.

D. Smith, D. Vier, T. Koschny, and C. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005).
[Crossref]

A. Starr, P. Rye, D. Smith, and S. Nemat-Nasser, “Fabrication and characterization of a negative-refractive-index composite metamaterial,” Phys. Rev. B 70(11), 113102 (2004).
[Crossref]

T. Koschny, P. Markoš, D. Smith, and C. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68(6), 065602 (2003).
[Crossref]

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[Crossref]

D. R. Smith, S. Schultz, P. Markoš, and C. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

Soukoulis, C.

R. Penciu, K. Aydin, M. Kafesaki, T. Koschny, E. Ozbay, E. Economou, and C. Soukoulis, “Multi-gap individual and coupled split-ring resonator structures,” Opt. Express 16(22), 18131–18144 (2008).
[Crossref]

D. Smith, D. Vier, T. Koschny, and C. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005).
[Crossref]

N. Katsarakis, T. Koschny, M. Kafesaki, E. Economou, E. Ozbay, and C. Soukoulis, “Left-and right-handed transmission peaks near the magnetic resonance frequency in composite metamaterials,” Phys. Rev. B 70(20), 201101 (2004).
[Crossref]

T. Koschny, P. Markoš, D. Smith, and C. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68(6), 065602 (2003).
[Crossref]

D. R. Smith, S. Schultz, P. Markoš, and C. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

Soukoulis, C. M.

Starr, A.

A. Starr, P. Rye, D. Smith, and S. Nemat-Nasser, “Fabrication and characterization of a negative-refractive-index composite metamaterial,” Phys. Rev. B 70(11), 113102 (2004).
[Crossref]

Sun, C.

V. Fokin, M. Ambati, C. Sun, and X. Zhang, “Method for retrieving effective properties of locally resonant acoustic metamaterials,” Phys. Rev. B 76(14), 144302 (2007).
[Crossref]

Szabó, Z.

Z. Szabó, G.-H. Park, R. Hedge, and E.-P. Li, “A unique extraction of metamaterial parameters based on Kramers–Kronig relationship,” IEEE Trans. Microwave Theory Tech. 58(10), 2646–2653 (2010).
[Crossref]

Varadan, V. V.

V. V. Varadan and R. Ro, “Unique retrieval of complex permittivity and permeability of dispersive materials from reflection and transmitted fields by enforcing causality,” IEEE Trans. Microwave Theory Tech. 55(10), 2224–2230 (2007).
[Crossref]

Vartiainen, E. M.

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers–Kronig Relations in Optical Materials Research, vol. 110 (Springer Science & Business Media, 2005).

Vier, D.

D. Smith, D. Vier, T. Koschny, and C. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005).
[Crossref]

Wan, G.

B. Fu, X. Ma, and G. Wan, “Retrieving the Constitutive Parameters of Metal Backed Radar Absorbing Material by Phase Unwrapping Method,” European Conference on Antennas and Propagation, 2018.

Wang, W.-C.

F.-J. Hsieh and W.-C. Wang, “Full extraction methods to retrieve effective refractive index and parameters of a bianisotropic metamaterial based on material dispersion models,” J. Appl. Phys. 112(6), 064907 (2012).
[Crossref]

Wegener, M.

Werner, D. H.

Wiltshire, M. C.

D. R. Smith, J. B. Pendry, and M. C. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[Crossref]

Wu, B.-I.

X. Chen, B.-I. Wu, J. A. Kong, and T. M. Grzegorczyk, “Retrieval of the effective constitutive parameters of bianisotropic metamaterials,” Phys. Rev. E 71(4), 046610 (2005).
[Crossref]

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70(1), 016608 (2004).
[Crossref]

Xiao, Z.

G. Lu, Z. Duan, H. Yin, Z. Xiao, and J. Zhang, “Determining the Effective Electromagnetic Parameters of Photonic Crystal by Phase Unwrapping and Denoising Method,” Int. J. Antenn. Propag. 2019, 1–10 (2019).
[Crossref]

Yeh, P.

G. T. Papadakis, P. Yeh, and H. A. Atwater, “Retrieval of material parameters for uniaxial metamaterials,” Phys. Rev. B 91(15), 155406 (2015).
[Crossref]

Yin, H.

G. Lu, Z. Duan, H. Yin, Z. Xiao, and J. Zhang, “Determining the Effective Electromagnetic Parameters of Photonic Crystal by Phase Unwrapping and Denoising Method,” Int. J. Antenn. Propag. 2019, 1–10 (2019).
[Crossref]

Yuan, F.

Z. Cao, F. Yuan, and L. Li, “An automated phase correction algorithm for retrieving permittivity and permeability of electromagnetic metamaterials,” AIP Adv. 4(6), 067115 (2014).
[Crossref]

Zhang, J.

G. Lu, Z. Duan, H. Yin, Z. Xiao, and J. Zhang, “Determining the Effective Electromagnetic Parameters of Photonic Crystal by Phase Unwrapping and Denoising Method,” Int. J. Antenn. Propag. 2019, 1–10 (2019).
[Crossref]

Zhang, X.

V. Fokin, M. Ambati, C. Sun, and X. Zhang, “Method for retrieving effective properties of locally resonant acoustic metamaterials,” Phys. Rev. B 76(14), 144302 (2007).
[Crossref]

AIP Adv. (1)

Z. Cao, F. Yuan, and L. Li, “An automated phase correction algorithm for retrieving permittivity and permeability of electromagnetic metamaterials,” AIP Adv. 4(6), 067115 (2014).
[Crossref]

IEEE Trans. Microwave Theory Tech. (3)

V. V. Varadan and R. Ro, “Unique retrieval of complex permittivity and permeability of dispersive materials from reflection and transmitted fields by enforcing causality,” IEEE Trans. Microwave Theory Tech. 55(10), 2224–2230 (2007).
[Crossref]

Y. Shi, Z.-Y. Li, K. Li, L. Li, and C.-H. Liang, “A retrieval method of effective electromagnetic parameters for inhomogeneous metamaterials,” IEEE Trans. Microwave Theory Tech. 65(4), 1160–1178 (2017).
[Crossref]

Z. Szabó, G.-H. Park, R. Hedge, and E.-P. Li, “A unique extraction of metamaterial parameters based on Kramers–Kronig relationship,” IEEE Trans. Microwave Theory Tech. 58(10), 2646–2653 (2010).
[Crossref]

Int. J. Antenn. Propag. (1)

G. Lu, Z. Duan, H. Yin, Z. Xiao, and J. Zhang, “Determining the Effective Electromagnetic Parameters of Photonic Crystal by Phase Unwrapping and Denoising Method,” Int. J. Antenn. Propag. 2019, 1–10 (2019).
[Crossref]

J. Appl. Phys. (1)

F.-J. Hsieh and W.-C. Wang, “Full extraction methods to retrieve effective refractive index and parameters of a bianisotropic metamaterial based on material dispersion models,” J. Appl. Phys. 112(6), 064907 (2012).
[Crossref]

Microw. Opt. Technol. Lett. (1)

Y. Shi, T. Hao, L. Li, and C. H. Liang, “An improved NRW method to extract electromagnetic parameters of metamaterials,” Microw. Opt. Technol. Lett. 58(3), 647–652 (2016).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. B (7)

N. Katsarakis, T. Koschny, M. Kafesaki, E. Economou, E. Ozbay, and C. Soukoulis, “Left-and right-handed transmission peaks near the magnetic resonance frequency in composite metamaterials,” Phys. Rev. B 70(20), 201101 (2004).
[Crossref]

K. Mnasri, A. Khrabustovskyi, M. Plum, and C. Rockstuhl, “Retrieving effective material parameters of metamaterials characterized by nonlocal constitutive relations,” Phys. Rev. B 99(3), 035442 (2019).
[Crossref]

G. T. Papadakis, P. Yeh, and H. A. Atwater, “Retrieval of material parameters for uniaxial metamaterials,” Phys. Rev. B 91(15), 155406 (2015).
[Crossref]

V. Fokin, M. Ambati, C. Sun, and X. Zhang, “Method for retrieving effective properties of locally resonant acoustic metamaterials,” Phys. Rev. B 76(14), 144302 (2007).
[Crossref]

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77(19), 195328 (2008).
[Crossref]

A. Starr, P. Rye, D. Smith, and S. Nemat-Nasser, “Fabrication and characterization of a negative-refractive-index composite metamaterial,” Phys. Rev. B 70(11), 113102 (2004).
[Crossref]

D. R. Smith, S. Schultz, P. Markoš, and C. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

Phys. Rev. E (4)

T. Koschny, P. Markoš, D. Smith, and C. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68(6), 065602 (2003).
[Crossref]

X. Chen, B.-I. Wu, J. A. Kong, and T. M. Grzegorczyk, “Retrieval of the effective constitutive parameters of bianisotropic metamaterials,” Phys. Rev. E 71(4), 046610 (2005).
[Crossref]

D. Smith, D. Vier, T. Koschny, and C. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005).
[Crossref]

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70(1), 016608 (2004).
[Crossref]

Prog. Electromagn. Res. (1)

U. C. Hasar, J. J. Barroso, C. Sabah, and Y. Kaya, “Resolving phase ambiguity in the inverse problem of reflection-only measurement methods,” Prog. Electromagn. Res. 129, 405–420 (2012).
[Crossref]

Science (1)

D. R. Smith, J. B. Pendry, and M. C. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[Crossref]

Waves in Random and Complex Media (1)

Y. Shi, Z.-Y. Li, L. Li, and C.-H. Liang, “An electromagnetic parameters extraction method for metamaterials based on phase unwrapping technique,” Waves in Random and Complex Media 26(4), 417–433 (2016).
[Crossref]

Other (2)

B. Fu, X. Ma, and G. Wan, “Retrieving the Constitutive Parameters of Metal Backed Radar Absorbing Material by Phase Unwrapping Method,” European Conference on Antennas and Propagation, 2018.

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers–Kronig Relations in Optical Materials Research, vol. 110 (Springer Science & Business Media, 2005).

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Figures (21)

Fig. 1.
Fig. 1. Metamaterial slab with normal incident plane wave.
Fig. 2.
Fig. 2. Permittivity of Drude model and permeability of Lorentz model.
Fig. 3.
Fig. 3. (a) Magnitude and (b) phase of reflection and transmission coefficients for $d = 40\; nm$.
Fig. 4.
Fig. 4. (a) Magnitude and (b) phase of reflection coefficients for $d = 200\; nm$.
Fig. 5.
Fig. 5. (a) Magnitude and (b) phase of reflection coefficients for $d = 400\; nm$.
Fig. 6.
Fig. 6. Wave impedance and refractive index of homogeneous slabs (d = 40 nm, d = 200 nm, and d = 400 nm): (a) The real part of complex wave impedance, (b) imaginary part of complex wave impedance, (c) imaginary part of refractive index and (d) real part of refractive index.
Fig. 7.
Fig. 7. Real part of the refractive index of the homogenous slab ($\textrm{d} = 40\textrm{ nm}$).
Fig. 8.
Fig. 8. (a) Real part of the refractive index of the homogenous slab ($\textrm{d} = 400\textrm{ nm}$). (b) Branch index of the homogenous slab ($\textrm{d} = 200\textrm{ nm}$).
Fig. 9.
Fig. 9. (a) Real part of the refractive index of the homogenous slab ($d = 400\;nm$). (b) Branch index of the homogenous slab ($d = 400\;nm$).
Fig. 10.
Fig. 10. The extracted and defined effective permittivity and permeability parameters of homogenous slab for $d = 40\;nm$ using (a) K–K method and (b) D-D method.
Fig. 11.
Fig. 11. The extracted and defined effective permittivity and permeability parameters of homogenous slab for $d = 200\;nm$ using (a) K–K method and (b) D-D method.
Fig. 12.
Fig. 12. The extracted and defined effective permittivity and permeability parameters of homogenous slab for $d = 400\;nm$ using (a) K–K method and (b) D-D method.
Fig. 13.
Fig. 13. Unit cell structure of metamaterial geometry consisting of metallic rods and SRRs separated by dielectric.
Fig. 14.
Fig. 14. Refractive index of one-unit cell SRR-rod (in $z$ direction).
Fig. 15.
Fig. 15. Branch index of one-unit cell SRR-rod.
Fig. 16.
Fig. 16. Refractive index of two-unit cell SRR-rod.
Fig. 17.
Fig. 17. Branch index of two-unit cell SRR-rod.
Fig. 18.
Fig. 18. Extracted effective (a) permittivity and (b) permeability of one-unit cell metamaterial, using the K-K method and the proposed method (D-D method).
Fig. 19.
Fig. 19. Extracted effective (a) permittivity and (b) permeability of two-unit cell metamaterial, using the K-K method and the proposed method (D-D method).
Fig. 20.
Fig. 20. Real refractive index of different frequency ranges and unit cell dimensions for K–K method and proposed method.
Fig. 21.
Fig. 21. Performance of proposed method and K–K method.

Tables (1)

Tables Icon

Table 1. Drude and Lorentz Parameters of Investigated Slabs.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

S 11 = ( z e f f 1 z e f f + 1 ) ( 1 e i 2 n e f f κ d ) 1 ( z e f f 1 z e f f + 1 ) 2 e i 2 n e f f κ d
S 21 = ( 1 ( z e f f 1 z e f f + 1 ) 2 ) e i n e f f κ d 1 ( z e f f 1 z e f f + 1 ) 2 e i 2 n e f f κ d
z e f f = ± ( 1 + S 11 ) 2 S 21 2 ( 1 S 11 ) 2 S 21 2
n e f f = i κ d l n ( S 11 1 S 21 ( z e f f 1 z e f f + 1 ) ) .
n = 1 κ d i m ( l n ( S 11 1 S 21 ( z 1 z + 1 ) ) ) + 2 m π κ d
n = 1 κ d R e ( l n ( S 11 1 S 21 ( z 1 z + 1 ) ) )
ε e f f = n e f f z e f f
μ e f f = n e f f z e f f .
n K K ( f ) = 1 + 2 π P 0 f n " ( f ) f 2 f 2 d f
m ( f ) = R o u n d [ ( n K K ( f ) n ( f ) ) κ d 2 π ] .
D ( f ) = d n ( f ) d f .
D ( f i ) = n ( f i ) n ( f i 1 ) f i f i 1 .
m i = { R o u n d [ ( n ( f i 1 ) n ( f i ) ) κ d 2 π ] f o r | D ( f i ) | = | q ( f i ) | 0 O t h e r w i s e
m = i = 0 N 1 m i .
q ( f i ) = 2 n ( f i 1 ) ( f i f i 1 ) .
ε e f f = ϵ ω p 2 ω 2 + i γ c ω .
μ e f f = μ ( μ s μ ) ω 0 2 ω 2 + i δ ω ω 0 2 .

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